Analytic Spectra of CMB Anisotropies and Polarization Generated by Scalar Perturbations in Synchronous Gauge

Le résumé fourni par la source

The temperature anisotropies and polarization of the cosmic microwave background (CMB) radiation not only serve as indispensable cosmological probes, but also provide a unique channel to detect relic gravitational waves (RGW) at very long wavelengths. Analytical studies of the anisotropies and polarization improve our understanding of various cosmic processes and help to separate the contribution of RGW from that of density perturbations. We present a detailed analytical calculation of CMB temperature anisotropies αk and polarization βk generated by scalar metric perturbations in synchronous gauge, parallel to our previous work with RGW as a generating source. This is realized primarily by an analytic time integration of Boltzmann's equation, yielding the closed forms of αk and βk. Approximations, such as the tight-coupling approximation for photons a priori to the recombination and the long-wavelength limit for scalar perturbations, are used. The residual gauge modes in scalar perturbations are analyzed and a proper joining condition of scalar perturbations at the radiation-matter equality is chosen, ensuring the continuity of energy perturbation. The resulting analytic expressions of the multipole moments of polarization aEl and of temperature anisotropies aTl are explicit functions of the scalar perturbations, recombination time, recombination width, photon-free streaming damping factor, baryon fraction, initial amplitude, primordial scalar spectral index and the running index. These results show that a longer recombination width yields higher amplitudes of polarization on large scales and more damping on small scales, and that a late recombination time shifts the peaks of to larger angular scales. Calculations show that aEl is generated in the presence of the quadrupole α2 of temperature anisotropies via scattering, both having similar structures and being smaller than the total aTl, which consists of the contributions from the monopole, dipole, quadrupole and Sachs–Wolfe terms as well. The origin of the two bumps in CEEl on large angular scales is found to be due to the time derivative of the monopole of temperature anisotropies. Furthermore, aEl together with aTl demonstrates explicitly that the peaks of CEEl and CTTl alternate in the l-space. These results substantially extend earlier analytic work. The analytic spectra agree with the numerical ones and with those observed by WMAP on large scales (l ≲ 500), but deviate considerably from the numerical results on smaller scales, showing the limitations of our approximate analytic calculations. Several possible improvements are pointed out for further studies.

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